The spectral moments of trees with given maximum degree

Let Δ≥3Δ≥3. Denote by Tn,ΔTn,Δ the set of all trees with nn vertices and maximum degree ΔΔ and by Tn,Δ∗ the set of all ΔΔ-trees with nn vertices. In this work, we first show that all ΔΔ-trees come before all trees in Tn,Δ∖Tn,Δ∗ in an SS-order and present a criterion for a ΔΔ-tree coming before another ΔΔ-tree in an SS-order. Also, we give the first ∑k=1⌊n−13⌋(⌊n−k−12⌋−k+1) graphs apart from a path, in an SS-order, of all trees with nn vertices.